Acta Mathematica

Convergence and divergence of formal CR mappings

Bernhard Lamel and Nordine Mir

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Let $M \subset \mathbb{C}^N$ be a generic real-analytic submanifold of finite type, $M' \subset \mathbb{C}^{N'}$ be a real-analytic set, and $p \in M$, where we assume that $N, N' \geqslant 2$. Let $H: (\mathbb{C}^N, p) \to \mathbb{C}^{N'}$ be a formal holomorphic mapping sending $M$ into $M'$, and let $\mathcal{E}_{M'}$ denote the set of points in $M'$ through which there passes a complex-analytic subvariety of positive dimension contained in $M'$. We show that, if $H$ does not send $M$ into $\mathcal{E}_{M'}$, then $H$ must be convergent. As a consequence, we derive the convergence of all formal holomorphic mappings when $M'$ does not contain any complex-analytic subvariety of positive dimension, answering by this a long-standing open question in the field. More generally, we establish necessary conditions for the existence of divergent formal maps, even when the target real-analytic set is foliated by complex-analytic subvarieties, allowing us to settle additional convergence problems such as e.g. for transversal formal maps between Levi-non-degenerate hypersurfaces and for formal maps with range in the tube over the light cone.


The authors were partially supported by the Qatar National Research Fund, NPRP project 7-511-1-098. The first author was also supported by the Austrian Science Fund FWF, Project I1776.

Article information

Acta Math., Volume 220, Number 2 (2018), 367-406.

Received: 26 September 2017
First available in Project Euclid: 19 June 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions 32H40: Boundary regularity of mappings 32V20: Analysis on CR manifolds 32V25: Extension of functions and other analytic objects from CR manifolds 32V40: Real submanifolds in complex manifolds

formal CR map convergence deformation complex-analytic subvariety


Lamel, Bernhard; Mir, Nordine. Convergence and divergence of formal CR mappings. Acta Math. 220 (2018), no. 2, 367--406. doi:10.4310/ACTA.2018.v220.n2.a5.

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